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Impedance equivalents calculator

Where “the numbers do not agree” comes from

The same impedance can be written two ways: as R and X in series or as R and X in parallel. A NanoVNA shows the first; handbooks on coils and tuned circuits often give the second. The numbers come out quite different, and that is where the feeling that the instrument is lying most often arises.

RsXsseries=RpXpparallel
The same impedance at one frequency: a NanoVNA shows the left form, the handbooks the right one
What to enter
Rs (Ω)
Xs (Ω)
Frequency (MHz)
System impedance Z₀ (Ω)
Equivalents
Series
Parallel
Q factor
Equivalent element
Series
Parallel
Match to Z₀
Reflection coefficient Γ
SWR
Return loss
Mismatch loss

📐 The formulas​

Series → parallel:

Q  = |Xs| / Rs
Rp = Rs × (1 + Q²)
Xp = Xs × (1 + 1/Q²)

Parallel → series:

Q  = Rp / |Xp|
Rs = Rp / (1 + Q²)
Xs = Xp / (1 + 1/Q²)

Match against Z₀:

Γ = (Z − Z₀) / (Z + Z₀)
SWR = (1 + |Γ|) / (1 − |Γ|)
RL[dB] = −20·log10|Γ|
Mismatch loss = −10·log10(1 − |Γ|²)

The equivalent component:

X > 0 → L = X / (2πf)
X < 0 → C = 1 / (2πf·|X|)
Where this calculator stops:
  • The equivalents agree at one frequency only. You cannot swap Rs for Rp in a circuit
  • The resistive part must be positive: a negative resistance is a generator, not a load
  • Return loss is always positive; a NanoVNA shows S11 with a minus sign, and it is the same number
  • Mismatch loss is not cable loss: the reflected power comes back, it does not vanish

🎯 In practice​

  • Working with a NanoVNA. The instrument beside you, the calculator in a tab, and every quantity to hand.
  • Checking against a datasheet that gives the parameters in the other form.
  • Designing a matching network: an L-network usually needs the parallel equivalent.
Practical advice:
  1. Check yourself by converting back: there and back should give the same number.
  2. The Q is the same in both forms — a handy invariant to check against.
  3. For Q > 10 the difference between Rs and Rp exceeds a hundredfold; that is where the “strange” numbers come from.
  4. An SWR of 10 with Rs = 10 Ω and Xs = 50 Ω is normal: reactance spoils a match more than the resistive part does.

© 2026 UR3PKI · CyberDev.Space · Content licensed under CC BY-NC-SA 4.0.

How to cite this calculator
UR3PKI. «Impedance equivalents calculator: series ↔ parallel». CyberDev.Space. https://cyberdev.space/en/radio/calculators/others/series_parallel (licence CC BY-NC-SA 4.0).

The licence lets you use this material freely, including in teaching materials, but only with attribution to the author and a link to the source.

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